Constructions of Plane Curves with Many Points

In this paper we investigate some plane curves with many points over Q, finite fields and cyclotomic fields. In a previous paper [4] the first two authors constructed a sequence of absolutely irreducible polynomials Pd(x, y) ∈ Z[x, y] of degree d having low height and many integral solutions to Pd(x, y) = 0. (The definition of these polynomials will be recalled in §4.) Here we construct further examples of polynomials of arbitrarily large degree d over Q with many rational zeros, improving the known record for the maximal number of rational zeros of a smooth polynomial in two variables over Q of given large degree. We also construct examples of two variable polynomials having the maximal theoretically possible number of zeros at roots of unity and over finite fields. Finally, we return to the polynomials Pd and show that for certain special values of d they have a few more zeros than were found there. Here is a more precise statements of the results obtained, with a few remarks about each one.